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Author Topic: Principles of The Logos Kairos Sumbola Sunthemata Summetria Theurgigical set  (Read 6478 times)
Description: Revised and Philosophical version of the axioms for the set FS
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« Reply #15 on: January 02, 2017, 11:56:51 AM »

Xxx its been a while since I looked at the principles of so called Mathematics, the distinct learnings of the Pythagorean Mathematikoi or Astrologers.
What they did is ideate topological perceptions and dynamics .
The idea, the foundation of ideation is the visual sense of any " thing" , or perception.
Typically this is expressed as " an idea- usually a visual one, or a visible form.


Thus a form in Grassmanns Formenlehre, and indeed in most of his peers minds was a thought Form, or an idea. Dedekind starts his analysis of number with this Greek idea .

Quite apart from the misinformation taught about the Stoikeia and about the Arithmoi and about Monas , I find the discussion of Book 5, allegedly the work of Eudoxus, is not only muted but confused.

Here I present a meditative translation of the first few lines of book 5. Here my initial identification of Skesis with the concept of a sketch is deepened by an etymological study of the word and particularly highlight the dichotomy with Skesis and kinesis, that is still and moving, static and dynamic, held and loosed to run free.


Quote
A Part is a magnitude of a magnitude ,
the lesser magnitude of the greater magnitude
When Laid down onto  it measures the greater
Thus a Multiple Form  is the greater form
Made of the lesser forms
When laid down it is measured by the lesser forms.

Logos is two homogenous magnitudes Which apply to determining the stage of development of a specific make of sketch form

Let Logos be Magnitudes" placed relative to each other" let's say,
which generating magnitudes we "form into Multiple Forms "  ( placed) relative to one another,letting be bigger and bigger magnitudes.

http://perseus.uchicago.edu/cgi-bin/philologic/getobject.pl?c.31:3:35.LSJ
And following entries. Peelikotees is an interrogative form, and therefore the noun form is nominalising a questioning activity.

In the case of "the  same  logoi"  magnitudes let be  declared :  the first placed near to the second and the third  placed  near to the fourth,
when the magnitude of the first and the third is  a dualled / duplicate multiple form  ,  to  the dualled/ duplicate  multiple form of the second and fourth ,
 That is :
Applied onto the picked out one of the singled out ones
applied onto any specifically numbered multiple form of it.
And the both exceeds or the both duals or the both  falls short Of the taken /picked  one (when ) laid down together.
Thus that case of  "the same logos" magnitudes existing shall be called AnaLogos.


 
[ the first statements of each book are definitions which are general. The general definitions make no sense without specific examples . The specific examples follow in the text and in point of fact are prior to the definitions! However the style is to pronounce the general form or principle drawn from experience like those given in the examples and then to give specific examples that reinforce and illustrate the general principle.
Thus any student moves from not knowing to knowing terminology that is meaningless to attaching meaning to the terminology to appreciating the thought pattern being illicitrd.
When this happens the student is initiated into the jargon of the senior level of discourse. A junior is not only evident by the use of unfamiliar terminology  but also by the lack of ability to demonstrate examples of the terminology, and finally by not demonstrating the thought pattern used in the discourse.

What if the thought pattern is " wrong" ? This can not be the case unless wrong means the pattern does not do what it is claimed it does. Thus if a logos is claimed to give eternal life then it is wrong but a logos is a particular pattern of 2 magnitudes which is used to generate multiple forms to determine the stage of development of a sketch of lines or flat shapes or solids , these latter 2 are dealt with in books 6 and 7
What I need to know : what are these dualled multiple forms
What is Skesis ?
What are the 2 homogenous magnitudes
When are 2 logoi declared the same?
What have to be taken together
What are both and what do they exceed,dual or fall short of ? ]

The first demonstration shows lines as magnitudes , it shows the generating magnitudes  and it shows the multiple forms generated by dualed multiple forming . In this case the dual is the same number of magnitudes forming each respective. Multiple form

Visually it is graspable, verbally it is convoluted! The ear , eye and thought are patterned to apprehend the same thing
A logos is 2 magnitudes treated as a single pattern
The relationships between magnitudes related in this way  is what is being explored .  The first demonstration is that multiple forms generated by the logos are dualled forms , that is the same or a dual number of magnitudes goes into each multiple form
If you know the relationship of one then you know the relationship of the other because they are dualled.
A logos is a pairing of any kind of magnitudes whatsoever? Strictly no , but the principles work for such Inhomogenous pairings.

As I meditate on this I recognise that Monas is not a number . It is anything that we wish to call a unit , a whole entire. Even if it may be constructed from smaller units like Arithmoi generally are, we subsume those into a new Monas or unit.
This is not the same as unity of parts, rather constituents of a unity may be inharmonious  either in behaviour, aesthetics or shape, but we by force of will and convention, even convenience and custom declare them to be a unit, and thusly unify them!!

The notion of a Metron is a secondary layer added to some topological form we clearly declare a unit. The Metron derives it's meaning from a song and dance process called katameetresee, in which we place down a unit while declaring or naming or singing a sound unique to that unit . But unique only in order in which we choose to proceed to sing !
Once laid down, the Arithmos is complete. But we may understand that this process of creating a perfect Arithmos require us to use a process of factorising.
We always factor a larger by a smaller topological form . Thus in a process where forms are at first not commensurable, being either too many Or too little by a unit , that is perisos( as opposed to artios that is perfect fit)  the Euclidean used the remainder to factor the whole thus they inverte the process by maintaining the rule: the smaller into the larger.
This inversion continues until a perfect fit is found. That smallest remainder that does that thing is now the unit of commensurability for the 2 initial topological forms.
But sometimes that remainder can not be found, and then the initial forms are said to be incommensurable.

The ratios and proportions of book 5 explore these relationships between topological forms by analysing it in terms of extensible magnitudes.

The layout for this exploration uses order( first second third and fourth etc) to keep track of the inversions in this process.
From this layout, much later a Persian geometer constructed the rules for a system we call fractions.

While fractions are purveyed as less than wholes, this is not the case. They are proportions and ratios of smaller units of one sort into larger units of another. Or more generally a lesser topological form into a greater topological form .
The ratios and proportions remain the same as laid out in books 5 to 7 of the Stoikeia

It is of great interest to note that from the outset these ratios and proportions deal with the dynamic process of stage development and how to depict it using recognised topological forms sketched in lines by the imagination of an artist ( Skesis)

The etymology of Skesis, and linear B  reveals the link to Aramaic, and the nearveasternnsemitic languages. In these languages the bilateral and trilateral root is important. The root "sk"  root is associated with the hand and all it's functions .
The main function is to hold
The secondary function is with the hold to manipulate what is held, and the third aspect is to depict outcomes of the first 2 processes.

Skazo means to slit by hand ( to loose something)
Then the sk words up to skesis have to do with holding , retaining by being near to hand , being hand made, hand tooled , hand drawn , held in position, shape or form So ribs are like fingers holding in the viscera, and also holding the form
The raft is a rough handmade form, binding together wood and Bouyant materials into a stable form .
All of these ideas feed into the generl notion of a skesis. Thus a Skesis is a fixed form made by hand . Normally the hand element is way in the background , but in this context where the performer is doing a lot of creative processes, the hand element comes to the fore

The idea of a sketch is seemingly tenuous, but it refers to the habit of geometers or rather Astrologers to sketch out the problem or situation either on paper or on the ground.

It is apparent now that Verh鄟tnisse is the German equivalent of Skesis, it being the everyway holded ness in space of some form or conception.

The Logos and the Analogos.
By the way I admit to formerly misinterpreting this section in the past. I hold my hand up as a hobbyist translator not a professional .
Please challenge anything you like , but without ad homiem logic!
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« Reply #16 on: January 19, 2017, 06:01:58 AM »

http://m.youtube.com/watch?v=3grjQB04cu8

Here is aclear attempt to translate Euclids/ Eudoxus Logis Analogos theory.
The definition is updated but clear. However it is brief and slightly misleading.

Se the previous post for a fuller explanation

The wot part is clear but the pattern magnitude of a magnitude is not expressed. Without this pattern homogeneity can not be explained. Two homogenous magnitudes are obtained from the same magnitude as parts.
The process of making or creating multiple forms is not expressed. Instead the word multiple is used without reference to this process.
The idea that a greater magnitude can be a multiple form created from a smaller form is not expressed.
The reason is the difficulty of self reference in the Greek language. Thus the smaller form is a part of the larger form and the larger form is a multiple form created from the smaller form . But the smaller form is also part of its multiple form generated from itself by a process of dialling or duplication.
Thus a multiple is formed or generated by a process of copying exactly or duplicating a form which is as a result a part of the multiple form and a lesser part of a greater magnitude or a lesser magnitude of a greater magnitude.

Now this is the meditative expression of the ideas, pattern of words and content of the first discourse in the introduction. Despite its economy of words the text is not meant to be hurried over, but meditated upon to extract all the relationships and processes bound within it.

The modern idea of an integer which means whole clearly clashes with the stated context of parts!  And the reason how it does that is through the misleading concept of whole numbers rather than Arithmoi which actually are defined in book 7 . The process of multiplication is also misleading because there is no process of multiplication except by dual long or duplication of a given form or part. .

The next idea is that of logos. Quite simply this is two parts from a larger form which are therefore homogenous. These parts can differ of be identical but the identical case is not used to define a logos. Two differing parts of the same form are used to define the word Logis!

It is almost self evident that one can say very little about one form or one of anything. In fact the main thing said or say able about one is that is is the singled out form of singled out forms.

Note the pattern again, established by magnitude of a magnitude.

One therefore is left until book 7 to be fully defined where Monas is clearly defined as a singled out form of a singled out collection of forms. It is left until then because by then it is quite clear to the discoursing and meditating student that one is a very complex idea and not simple at all!
Monas has an incredible discourse devoted to it by the Neopythagoreans.

So the logos or the word is that two parts are laced next to each other in an unspecified way and those two parts are to be used to capture the development of a dynamic form . This dynamic form is developed by generating duplicates of the two initial or generating forms and making multiple forms which are therefore larger than or exceed the original homogenous pair of parts. These multiple forms are also placed next to each other . Consequently these multiple form pairs are also called logos, because the same word can be said about them.

We now concentrate on the next idea which is Analogos  again this is a pair of logoi considered together. The pair of logoi are referenced by their parts. The first and second part form one Logos and the third and fourth part form a second Logos

We have just discussed how these logoi may be generated and the idea now is to define the same logoi by considering pairs of logoi. When we consider pairs of logoi this is defined as Analogos if and onl if the duplication of the first part gives the multiple form in the third part, and a duplicate duplication process on the second part gives the multiple firm in the fourth part.

Analogos is thus about different logoi being declared the same or identical despite the fact that the parts are not the same! The two parts or both parts of the Analogos can be greater or exceed, or they can just dual, that is be an exact duplicate, or they can be less than the logos.
the logos is thus parts the first and second, while ana( new or again) logos are partscthe third and fourth , and it is only thusly called if their is this duplicate duplicating process between the two or duplicated logoi!

This heavy utilising of the duplication idea is an induction into the inductive reasoning and synthesis method of the pythagoreans. Because every statement of an idea or a process is duplicated at a different level, the next level up, the synthesis and arguments are irrefutable.

And yet Atistotle attacked the process of synthesis on precisely this point. He tried to refute the notion of two or two ess or duality. I was not convinced by his refutation nor his argument against dual processes of synthesis or indeed against Pythagorean induction in general. It seemed to me rather that Aristotle , brilliant as he was, had do ewhat more to learn of the Pythagorean thought form, but he fixated more on Plato than on the teachings of the Pythagorean Mathdmatikoi of which Euclid and Eudoxys are both recognised members who have achieved that  status. .
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« Reply #17 on: January 19, 2017, 07:26:01 AM »

The Ana Logos label is worth u dear standing a bit more.

http://m.youtube.com/watch?v=GnrCJiDXu2E

First we have two parts of a form used to generate different multiple forms of those parts.
The parts are paired in an unspecified manner exce
T they are near to each other, and when they are paired in this way n expression is loosed out of the mouth and mind! Thus the simplest word, reference or description or comparison is two parts placed or conceived together. We could say "this pair" or" this relationship of two parts.". Whatefpver we say it is ubstantially more than we cnnsay of a part of a form.
So the idea of defining or declaring two logoi as the same or identical or the self reveals the complexity of self referencing onthe Greek. The two logoi, the pair of logoi are declared self referencing  when they actually reference each other! They do this when the multiple forms of the oarts of the logoi are generated from each other by a duplicating process the duplicates each part to a specified multiple form , and that specified multiple form is duplicated for the picked out or ingles out fist and third magnitudes and second and fourth magnitudes.

So analogy is not based on omparing one homogenous magnitude or fom to mother, but on comparing a homogenous pair of forms to another pair which demonstrate a self referencing property. . We call this self reference similarity or proportion the Greek word is analogous.
Two lines can not be of themselves analohous, even if one is generated as a multiple form of the other by duplicating pa part. The idea is to ompare - pir of parts and then compare another pair nd thn to determine if the logoi are nous.

The next definition defines when one logos is greater than the second logo.

It should be noted that the tutor refers to the confusing nature of the language . English avoids many self referencing verbs or ideas while it is a common feature of ncirnt near eastern languages.  The simpler clearer language preferred by English speakers is actually less rich in implications and applications. . The meditative discourse is impoverished b these impolitic translations and the student is further disadvantaged by not being challenged to express clearly what they are experiencing  and to sketch or draw their Ideas

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« Reply #18 on: March 01, 2017, 03:06:08 PM »

http://m.youtube.com/watch?v=9qEST41kG6Q
Somewhere in the Fractal foundations of mathematics I discuss Eulers equation for geometrical firms.
V+F-E=2. This I  related to points , lines surfaces and volumes to show that a kind of expression captures an invariant truth about rectilinear forms, and forms the basis for generalising into higher order or mor complex structures.

This is the same idea explained here but presented as a general concept called a vector field.

Norman Wildberger presents a powerful series showing Eulers formula to be topologically relevant , and especially when the lines are viewed as connections in a network. Here the topological constraint is area and volume.
2 surprising ideas are mentioned, dimension and orientation.

http://m.youtube.com/watch?v=babedyM9dXg

I believe Normans introduction to linear spaces is the most accessible general presentation. The concepts of span, linear combinations etc are carefully based on prior developments. The concepts of multisession, vexes, maxes etc all have clear if unfamiliar definitions. In addition, Norman does not shy away from the mathmythics and analogous thinking patterns that go into constructing a mathematical concept like dimension.

I might add that I do not agree with some terminological niceties like linear instead of lineal, but the exposition is consonant with Grassmanns original ideas.

So we can get an idea of dimension as a convenient way of synthesising the lines in a 3d space, a 2d space and 1d space. But then because we want to generalise from a specific case to analogous cases we drop the saciometrically interpretation and concentrate on the symbolic arithmetic, or Algebra

Now Hermanns Grassmann recognised this as a articulately Hegelian dialectical approach , and his book, the Ausdehnungslehre is written from the general to the specific instances, as if this is how we think, hw we derive from these general thought patterns specific creative solutions.

But Hermanns explains this is not the case. He came to this presentation after much trial and error , much inspiration that left major gaps that needed time to fill in, and he had so little time!!

The story is declared in the thread The Theory of Stretchy Thingys!!

I mention it because few find the purity of this presentation( Fōrderung) easy to deal with. And the concept of orientation, a fundamental spaciometric concept comes in right at the beginning after a general combinatorics discussion requires it.
Norman introduces right at the beginning as a property of the natural numbers  when written as multisets. The arrangement of marks on the page demand sequence, spatial orientation , structural form etc.. As spaciometric entities, our proprioceotion demands the concept of orientation .

But here s the rub: Hermanns used the concept of direction rather than pure orientation, here Macdonald uses a non spatially specific concept of orientation except in the scalers where he has to specify an order relation as an orientation.
Hermanns took his primitives to be continuos line segments,Strecken, and thus properties of orientation, direction and magnitude and quantum naturally follow, or are inherent in the primitive.

However in his presentation he starts with the undefined primitive of a dynamic point.there is an interplay between this creative dynamic primitive and the static reference point defined using it in a tautological statement early on in his presentation.. It is this dynamic creative point that gives ' direction' through its travel to a created concept called a line. The line is created by this dynamic primitive entity.

In "reality" it is a fancy for our mutable perceptive abilities, in that I may perceive such a point as dynamically creating a path called a line, even where no such physical dynamic palpably exists.

So in this regard, orientation is used to invoke the direction of this creative point, and therefore it fails for a solid , returning to a single continuous line that traverses throughout the space!

The dimension of 1 for a solid is because one set of vectors describes every reference point. We could therefore creat every reference poit by a continuous line that travels to and through every point.
Hilbert showed that we can logically come to this conclusion by a limit argument. Of course you need to be convinced such infinite actions make sense! But it is possible to do this fr any finite set of rational reference points.it would have to be by this line that " orientation" of a solid is defined, and as a starting point is arbitrary, so will any orientation be!
The use of rotation to define orientation sets up some interesting problems. Essentially we could spiral around a diagonal like an onion in layers, and that reveals the essential rotational dy amic we absolutely need to make sense of our space!

It was this freedom of rotation that baffled Hermanns, but which drove Hamilton to a brute force solution for 3 dimensions we call the Quaternions.
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« Reply #19 on: March 04, 2017, 01:34:34 PM »

It dawned on me during a bus ride that linear ombinations, better general combinations, are not restricted to oriented li e segments . In fact I was thinking about the minimum basis to reference motion, and realised that 3 dimensions are fine for point referenced motion along some line, but motion in a general p,and requires some serious constraints.
I could use on st of orthogonal reference frames to give position. Then a different set to give rotation, consisting of trigonometric lines. But the rotation of a sphere is not reducible to one set of orthogonal trigonometric linesc because those lines would be oriented  in the position frame and give a different behaviour for each orientation. The orientation does not affect the combination, only the terms combined, and those terms are not reducible to others by combination . So in general to reference motion we need as many orthogonal trigonometric sets as necessary to describe the motion. This would be a general combination and so we would have at least 9 dimensions for any motion and probably more !
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« Reply #20 on: March 06, 2017, 12:19:24 PM »

http://m.youtube.com/watch?v=8vBfTyBPu-4

I thought I might try to grasp the notions of co and contravariance.

http://m.youtube.com/watch?v=AKPZkHvqTao

As usual, the mathmythics fogs up the view.
First consider Thales theorem ; in a semi circle one always sub tends a right angle at the perimeter. . The straight lines that are subtending, that is holding below, actually hold the diameter below the semi circular perimeter, from a point on it. These lines are the sides or limbs of a right angled triangle that stands on the diameter.

So now consider a full circle crossed by a diameter. I can find 2 points one in either semi perimeter such that the lines at one end of the diameter are at a given angle between them . Such a pair of lines may be taken as the reference frame for a plane coordinate system.

When such a pair of lines are used they clearly vary by the perimeter of this circle, and the diameter can be seen to project orthogonally onto these lines in each semi circle from the opposing end. This dynamic is called co variance of the reference frame lines.

The lines do not in fact vary together as co implies, they vary independently depending on the desire of the observer.  So the term is immediately misleading.

The lines are on strained by the diameter and the perimeter of the circle to relate by Thales theorem

However, if we concentrate on the fact the vertical projection onto the reference lines ar both the diameter and a Cosine product then we could call them covariants, either as a shortened " Cosine variant"  or a co variant as in fellow  type or kind of variant of the diameter
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« Reply #21 on: March 06, 2017, 09:22:32 PM »

The contra variant uses parallel projection as the resolving process. In this case the controlling figure is a arallelogram with a fixed diameter while the other diameter is allowed to freely extend as the lines remain parallel.
We may take one end of this fixed diameter as the place to fix a reference frame. In this case the parallel projection onto these lines now vary in opposing ways with the angle between them . In this case the projected lines cut off varying magnitudes  of the reference lines, but in a way that is co varying, dependent on each other by the angle between them!  The contra idea comes only from the observation that they extend in an opposing manner while not truly opposite extensions.

The introduction of the tensor matrix us interesting. These change if basis processes can be encapsulated in an array. The array sets out systematically on the page the factors of a product.

The idea of setting the process out in this way is ascribed among others to Cayley, but in fact Grassmann spent most of the Ausdehnunslehre specially setting out these product sums. The array or aerie was key to his conception even if he did not create a new tabular presentation of the factors.

It is this tabular arrangement that was at first called a table and then a Matrix, and to a certain extent zest Vainant contributed to this key to understanding the then modern mathematical processes.

We must however understand that it was Peano who took Hermanns Grassmans work to the format we mysteriously cLlmTensirs! There is no mystery, the Italian Tensor means to stretch just as the German Ausdehnungs does. So the Tensor theory relied heavily on the summation convention or the element with subscripts for row and column inventions. . The Cayley table format gave a specific table to wrk with rather than just a symbolic element with row column identifiers.

Certain products could be easily distinguished by this notational device, however, and whole matrices could be referred to elegantly. The distinction between circular based referee frames and parallelogram based reference frames became so useful that the superscript notation was devised by Ricci and Levi, and taught to Einstein.

These arrays were identified as algebraic in themselves and this enabled Ricci to depict complex combinations of terms to produce products very elegantly. The matrix that enabled certain equations to be expressed legantly providing they were affine projections or circular projections were called by the Ricci Levi school Tensors. They represented 2 types of extension or extensive magnitudes; the trigonometric and the affine.  The projective extension came later and possibly the most accessible proponent of that is NJ Wildberger.

http://m.youtube.com/watch?v=6MstJEAlNFs

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« Reply #22 on: March 08, 2017, 12:43:06 PM »

I have been attempting to understand trochoidal dynamics.
Hang on , I have tried to expand my thinking to free it from the tyranny of the straight line!
Forgive the poetry.
I have struggled for years to apprehend what mathematics and mathematical physics and chemistry is about.
Finally I accepted thebGrassmanns approach, a constructivist approach. . To cut a long introduction short  searching for Grassmann in YouTube brought me into contact with NJWildberger.

Now I have used his research nd understanding to better grasp the Grassmanns group and ring theoretical combinatorial approach. But then it became clear they were expounding on the Pythagorean school of thought.

My interest was then piqued by the fundamentally dynamic basis provided by the circle or sphere. Then I found out that 琀sted had found experimental evidence for a circular force dynamic being inherent in space. Then I recently found the Boscovich exposition of inherent force expounding on Newtonial principles.

So the Benoit Mandelbrot Fractal topology revolution gave me a computational dynamic for rotational dynamics . Later I came across Kegan J Brills trochoidal spiral surfaces  and William Shank and his Trochoidal  applications along with TerrybGintz  fractal generators.

Some original research I to the imaginaries convinced me that the mystery was man made. The quarter arc magnitude , while topologically familiar was void in the algebraic or symbolic arithmetic setting.

It has taken a while to grasp how a topological quarter arc is combined on the page, in space and in space-time.

So I found a Will Shank app called TroTorted very suggestive for representing a Boscovichian pressure dynamic.

I got excited enough to try to depict the Hydrogen datac. This revealed that Trotorted was limited by its design
Finally I was driven to  another Will Shank app called Circa. , which was clearly just planar. However the addition of a dynamic rotation on screen meant that visually at least I could trick the eye into appreciating a 3d aspect to the trochoidal forms.

In coming to this arrangement of the phase scrubbers in Circa I realised that it was an implementation of NUWildbergers  ISpan technology, with dynamics added.
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« Reply #23 on: March 08, 2017, 01:08:31 PM »

http://m.youtube.com/watch?=SZmlo49D1NY
The Mset data set basis to integers or whole form conceptions
While the anti mark is useful it is better to think of a mirror form
While a mirror image has no reality a mirror form is a partial negative that cancels out part of the real form.

Thus perpendicular reflected dimensions are negatives but those that are arallelogram to the mirror are unaffected .
The best negative is the hole that snugly fits a whole cut out of it!
I discuss ths shunyasutra duality elsewhere . Here Normal usefully calls it a tracheotomy. Nothing is in fact something in our perceptions and it is the absence of comparable differences in perceived form .

The forms I meditate on are quarter arcs and spheroidal octants
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« Reply #24 on: March 08, 2017, 11:48:58 PM »

http://m.youtube.com/watch?v=6XghF70fqkY
Here Norman introduces the bi vector and trivet or so called in linear algebra.
The product of 2 line segments was and is an idea that requires explanation.
The basic explanation is design. A product, unlike addition and subtraction is a designed notation or label.

Addition and subtraction are natural space like dynamics. We can consider aggregation and did aggregation for example. But multiplication we do not see dynamically or naturally.

What we do see is duplication or copying an identity, sometimes not exactly identically. So we see multiplicatr for es. Almost the same if not exactly the same, produced by duplication in sace- like ways.

Logically this was untenable, one can not prove identicality, it is agreed or assented to.

To build a logically sound system Grassmann took indisputable dy amicus and then based everything on those dynamics by simply analogising the dynamic carefully. It is explained in abstract detail in the induction, but in the first chapter it is set out as an inductive synthesis!
Thus the first step is set and the following steps are synthesised by a process that inductively uses the prior step to get the current step. Then the process is applied again to the current step to get the next step. And so it proceeds.

One of the consequences of this was that if a line a labels the points between A and B, the beginning and ending points of a dynamic thenm-a was set to lable the dynamic from B to A .

The inductive method  for consistency meant that ab the continuous form that starts a line a and finishes at line a' by parallel transport in the direction of line b.  Notationally then and inductively Hermanns insisted we start with a and finish with b thusba not only had to be the the reverse of the previous synthesis process it also had to take the negative sign when corresponded or equated to the geometrical form.

What made it imperative was the strict application of the inductive process itself.

There are definitional ways to condpstruct an algebra and Norman and others do this freely, but as far as Hermanns was concerned this was too arbitrary, and not logically defensible, whereas no one can impeach an inductive synthesis, except to say how weird it is to the common sense! But it is precisely the common sense that fails when one ventures into higher and higher inductive steps or stages or dimensions.
The inductive process gives certainty by giving invariant process no matter what the outcome.

It is also an amazing discipline of construction or synthesis, so much becomes consistently manageable , consistently structured and confidently applicable.

Hermanns develops the theory by using elements and confirms the solidity of the labelling pattern. However he designs different products as he develops his expertise . The spreading apart product, which is based on inductively designed is accompanied by the colliding together product which is designed to use the perpendicular projection of the right triangle . Then he designs a quotient product which enables him to do rational and inverse computations algebraically, and thus represent division as analytically finding the factors in a multiplication process.

There are other products he designs some not as useful as others , and by 1877 he was able to demonstrate how the methods he developed designed the Quaternion algebra.

The doctrine he espoused was a system of expert analogies and intuitions based on rigorous research in combinatorial processes that have their basis in natural dynamics . The essential creative entity was and is the dynamic creative point that creates the line, whether curved or straight.

The Erlangen project  of Felix Klein, the transformational Geometry he espoused is in man ways a take on the apprehension of dynamics required for the modern world, the Leibnizian analysis of the way forward for Natural philosophy and physics and a part of the appreciation of the failure of the mathematics or geometry of the time.

The reforms in Prussia led to a Prussian Renaissance that Gauss tried to steer through his Protog Riemann, but in fact he ignored the Genius of the Grassmanns who were nearer to the solution than any high academician
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« Reply #25 on: March 11, 2017, 08:47:45 AM »

http://m.youtube.com/watch?v=rz8A5l_yn34

The coining of the ideas of covariance and contravariance take me right back tomHermann Grassmann via Peano. The contravariant idea is what Hermnn called the Stepping apart or spreading out product, and of course the covariant product is the colliding together product .
The covariant remarks upon the coming together of the variation, typically through vertical projection or a dropping of a perpendicular.  This is such a general rule that one hardly bothers to find out where it comes from. Certainly in Eudoxus and Euclids time the quarter turn was more useful than a general perpendicular   Except where computation was required, then Pythagoras theorem was indisputably necessary and the adjacent limbs of a right triangle were thought of together , ans a varying together. The actual calculated variation took centuries to tabulate to a useful degree of accuracy. These measures are as valid in spaceship navigation as in seafaring navigation!

The contravariant  behaviour involves the parallelogram. As many know cyclic polygons exist within circular boundaries, but mus vary covariant ly, that is as the right triangle varies, but once a constraint regarding the diameter of. Parallelogram is applied it becomes clear that the sides  have to vary in opposite ways . The angles at the diameter increase while the angles at the other non fixed diameter de tease. . That diameter extends while the other reminds constant . The constraint of parallelism ensures this contravariance in the angles. Of the vertices of the parallelogram.

The astute observer will not that the the vertices of a right triangle also contravary, but on suchnacwaybthatbthe non diameter angle is always 90, thus theybcoopperatebto implement one another.  One another, to complete on another in that sense.

In the video you see how a constant perimeter parlllogram rotates within the constant trochoidal,dynamic , but as the perimeter busies so does the size of the Trochoid. . Thus contravariant and covariant measures serve to describe trochoidal dynamicsv
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« Reply #26 on: May 30, 2017, 12:34:17 PM »

http://m.youtube.com/watch?v=e17J0mOhelQ

In making this presentation Norman once again reveals the power of The Grassmanns et. al. Work on lineal algebra. The dot product, the cross product, the matrix form and the determinant are all carefully placed to unify the rational trigonometry he developed and highlights.

However, much of this work is based on traditional mathematical ideas at primary and secondary level including the derivation of the Cosine formula. What is sidestepped here is the use of or need for trig tables because he is not interested in the arc "length" or rather arc segment associated in these tables.

It has taken me a while to apprehend where my maths education was deficient and actually mislead me from the use and meaning of the trig tables, which were developed for astrologers who sought to place stars on various spheres in the night sky in such a way as to match their observations.

The tables associate arc length to Pythagoras theorem or in terms of forms the arc segments arc sectors, chord segments etc to Thales theorem of the right triangle in a semi circle, through the ratio of squares on the right triangles in a given standard circle.

Reducing these line and arc segments to lengths creates a problem of measurement, whereas considering them as dynamic extensions means that a rolling disc can connect the arc extension to the diameter of any given disc. Thus essentially every formulation for a straight line segment can be factored by Pi to apply to an arc extension.
The dot product and cross product of an arc segment with a general other arc segment or the cross product of a matrix of arc segments can thus be computationally and algebraically derived, without the use of tables.

Tables become useful when a specific circle or sphere is given so that actual approximate measures can be computed.

Thus for example given a wheel of a given diameter tables enable one to calculate the movement of the centre along the diametr for a given radian arc length. In fact the radian arc length is designed to make that calculation particularly straight forward.

Given a fixed centre one can measure the circular velocity along a given arc segment given the radius and the standard radian measures.

The general spiral motion therefore is a dynamic combination of circular arc vectors and line vectors, and it is necessarily approached by bitwise dynamic approximations familiar in the ideas of calculus, but readily generated by a fractal generator!

It is this apprehension of the dynamic application of forms in a fractal generator by means of the inductive steps that liberates the presentation of calculus as a scripted set of notations into a dynamic accessible plaything which does so much more than calculate numbers either as areas or rates of change, but through the distance estimation, the surface plotting, the colour cycling the constraints produces in a modern computer a viewable form,

The issue is to take that process and understand the product and see if it gives insight into our utilisation of natural processes in technology and apprehending those natural processes in a rational way.

And yet we must avoid the conceit that our meddling models are in anyway fundamental laws of our universe, and humbly confine ourselves to developing some expertise that may have some practical use in our society.

The reliance on the mathesis of the imaginaries has no place in a modern apprehension of measure and calculation . It is firmly and definitely a measure of circular arc extension starting with the natural semi perimeter of a circle but more usually based on the quarter arc, both related by the right angle they contain.
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« Reply #27 on: July 26, 2017, 11:52:54 AM »

Circle theorems based on " angles" are really Based on Arcs
http://m.youtube.com/watch?v=opXl43e254Q
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« Reply #28 on: August 29, 2017, 11:19:50 AM »

I am always fascinated by the problem of trisecting a general angle. The reason being when I was a child I was introduced to engineering drawing one cold autumn day, and shown how to draw a sine curve from a circle .

I grasped it fom a co ordinate perspective but not from a dynamic one. How could you draw. Circle moving ?

The ideas are straightforward, but I had obscured important steps from my minds eye for so long it seemed n impossible task,

I knew it was not but I could not remember why.

These videos helped me remember , and also suggested another approximate method to trisecting the general angle, by using a cycloid!

Circle theorems and constructions are fundamental to trochoidal dynamics and kinematics .

<a href="http://www.youtube.com/v/EHMZkYhOX0E&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/EHMZkYhOX0E&rel=1&fs=1&hd=1</a>

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« Reply #29 on: September 06, 2017, 09:46:27 AM »

<a href="http://www.youtube.com/v/L24GzTaOll0&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/L24GzTaOll0&rel=1&fs=1&hd=1</a>

http://m.youtube.com/watch?v=L24GzTaOll0

This has caused a bit of a sensation in the news, but it is August when news is traditionally a bit less seriously reported.
It's worth understanding that the Arithmoi were regular geometrical forms not numerals. So a measure was counted , and different measures were utilised to count. . For construction areal measures were utilised for accuracy and length for example was not considered apart from a standard figure. In this case the Square was the standard and lengths and areas and corners measured by it in various ways.

The measurement of tri corners or trigonometry was done by ratios of squares . The squares were divided into sub squares of the standard square using 60 or some power of 60

60 was chosen because of its factor tabulation.
Factors 1,2,3,4 ,5,6,10 are immediately available.

Using 120(2x60) , 180(3x60) , adds factors 8 , then 9. 420( 7x60) adds factor 7.

So all the factors 1 through 10 are available through on multiple of 60
 Exact calculations are therefore discoverable for many fractional or sub unit measure.

We have been taught to normalise our tri corner ratios on a fixed radius, they normalised on a fixed square side up to a maximal diagonal.they avoided that ratio precisely because it was irrational. But they went on to calculate approximations for that ratio to a useful level of accuracy.

The exactness of there counts depended on what multiple of 60 they used. For example 2x3x7x11x13x60 gives them all the factors up to 16 . You will note the " prime " factors are what extend utility of the base 60 factorisation tables.
Within these tables suare counts and cube counts can be identified.

The factors 3,4,5 are significant because the Pythagorean triple can be expressed as (4-1),4,(4+1)  and that pattern is found in their general process of finding squared diagonals.

Proposition 1:14 explains the Egyptian  Greek form of this circular relationship.

Thus we see the Arithmoi a pattern of forms represented by a pattern of pressed orms  enabling the tables of Gometry. Pythagoras is said to have pointed this out: without the Arithmoi no one can perform geometry!

The factors of 60 involved in this triple  enable us to find 45,60,75 and 60,80,100 very readily. The 5,12,17 triples require us to extend the prime factors  that multiply 60 to include 17, we can hunt for triples using this kind of process. Prime numbers also rapidly become apparent, and their use as defining new factorisation tables evident. Thus the name proto or first  in the Greek. They are the first Arithmos of a new factorisation paradigm.
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May a trochoid of 去逸 iteratively entrain your Logos Response transforming into iridescent fractals of orgasmic delight and joy, with kindness, peace and gratitude at all scales within your experience. I beg of you to enrich others as you have been enriched, in vorticose pulsations of extravagance!
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